↓I’m assuming this won’t make much sense unless you’ve read the previous short post.↓
Let me lay out the methodology briefly.
Do you remember the moment you first became able to add numbers? I don’t.
How many readers truly understand why 2 + 3 = 5?
Most of us just remember that when the symbols “2” and “3” have a “+” between them and an “=” at the end, the answer is “5”.
I don’t remember ever being taught that it’s actually a procedural structure like:
(1+1+1) + (1+1) = 5Kids who struggle with math often aren’t shown these procedures at all—they’re forced into giving a “snapshot answer”.
When they see the string “2+3=”, aren’t they just outputting whatever has the highest likelihood in memory?
Is that even calculation?
But that’s not the main topic today, so I’ll skip that.
Anyway, with the “=ⁿ” structure, you can tell whether someone answered by “snapshot” or by actually running the procedure.
You just ask them to break it down with =ⁿ. Nothing difficult about that (at least from my perspective).
Snapshot = only the result
Procedure = the internal path
You don’t need to expand everything fully, but you can make the calculation process visible, and it becomes far easier for someone to explain their own reasoning.
For example:
2×3×4
=³ (2×3)×4
=² 6×4
= 24This way, you can see what kind of procedural strategy they used.
It’s kinder to kids whose logical skills aren’t fully developed yet.
I won’t cover multiplication or division in this post, so the example above is just for illustration.
And as for why it’s not “=¹”—I mentioned this in the previous post, but basically I don’t think the extra notation is necessary.
If other people eventually use this, the notation will drift and settle somewhere on its own anyway.
Let’s start with subtraction.
regular expression:
5 − 4 = 1A =ⁿ expression (a Δⁿ expression):
It’s easier for kids if it’s shown visually.
5−4 =²
(1+1+1+1+1) ← the “actual pieces” a child can understand
−
(1+1+1+1) ← visualizing what’s being taken away
= 1
※ This isn’t long-form subtraction; it’s just a collection of sticks laid out.
Not every part of this notation is “Δ”—what matters here is simply that =² is being used to expose the procedure.
With this kind of visual representation, you can teach it as “the top and bottom collide and disappear”.
If you use a horizontal bar in the middle, you can tell the kid “use your eraser and delete the matching number of sticks”.
Concrete actions are easier to learn than abstract operations.
In the end, they just count the remaining sticks.
Of course, this visual form is only a scaffold for understanding.
As kids grow older, their math concepts will update gradually—just like how the names we use for cars change over time.
It’s basically a temporary childhood buzzword.
If you remove the parentheses and plus signs and keep only the sticks, it becomes this.
You could say this is an even lower level of abstraction.
The idea is that when sticks collide, they cancel out.
Since this isn’t long-form subtraction, you might need to arrange it in whatever way is easiest to see.
(They’re not “carried digits”; they’re just sticks laid out, like lllll.)
11111
-
1111
=² |×|×|×|×|1|
= 1Next is addition, though in most cases you don’t even need diagrams for it.
If a child can raise their fingers, they can count up to ten.
All they need to remember is the relationship between the written numeral and the number of fingers.
If one finger on one hand is raised, that’s 1.
If two fingers are raised, that’s 2.
For “1 + 1 =”, if the child raises one finger on each hand, they can see that there are two fingers in total.
If they already understand that “two fingers on one hand = 2”, then “one finger on each hand” should also naturally register as 2.
This same finger-play works for subtraction as well.
For “1 − 1 =”, have the child bump the two raised fingers together and fold one down.
As long as they understand that a closed fist represents 0, it should be fine.
Addition is “the operation of increasing fingers”,
and subtraction is “the operation of folding fingers down”.
Both can be taught as transformations of the same underlying finger state.
=ⁿ should be thought of as “a mode that temporarily opens up the intermediate steps of a procedure”.
A Δⁿ expression is basically a way to turn an invisible process into a visible action.
・I opened a related question on Hacker News:
If you want to share how you learned arithmetic,
or just read how others learned, you can find it here.
Diary:
Feels like there are still gaps in the explanation—like carrying in addition, or other bits.
If something’s missing, maybe someone will point it out.
As for why it’s Δ: it’s because of “dragon”, obviously.
But that story is too long for today.
For now, just take Δ to mean something like,
“show me what the internal procedure actually looks like”.
It’s basically an instance of A’tell.
Formally speaking, this Δ-style demand isn’t a problem at all,
but in everyday conversation it would probably come across as rude.
If I say that “Δⁿ sort” is foundational knowledge for becoming a “dragon”,
you’ll get the vibe.
The next post will cover multiplication and division.
The one after that might lay out a roadmap for the order in which kids should learn arithmetic concepts—or maybe it won’t.
I might also throw in some rougher techniques for using Δⁿ sort in actual calculations.
Anyway, that’s enough for now. I’m heading to my part-time job.